A031435 -id:A031435 - cp's OEIS frontend

A003320 a(n) = max_{k=0..n} k^(n-k).

1, 1, 1, 2, 4, 9, 27, 81, 256, 1024, 4096, 16384, 78125, 390625, 1953125, 10077696, 60466176, 362797056, 2176782336, 13841287201, 96889010407, 678223072849, 4747561509943, 35184372088832, 281474976710656, 2251799813685248

Offset: 0

N. J. A. Sloane, R. K. Guy

For n > 0, a(n+1) = largest term of row n in triangles A051129 and A247358. - Reinhard Zumkeller, Sep 14 2014

			a(5) = max(5^0, 4^1, 3^2, 2^3, 1^4, 0^5) = max(1,4,9,8,1,0) = 9.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 231.

Seiichi Manyama, Table of n, a(n) for n = 0..599 (terms 0..100 from T. D. Noe).
D. Easdown, Minimal faithful permutation and transformation representations of groups and semigroups, Contemporary Math. (1992), Vol. 131 (Part 3), 75-84.
R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
I. Tomescu, Excerpts from "Introducese in Combinatorica" (1972), pp. 230-1, 44-5, 128-9. (Annotated scanned copy)

Cf. A003992, A031435, A056155.

a003320 n = maximum $ zipWith (^) [0 .. n] [n, n-1 ..]
-- Reinhard Zumkeller, Jun 24 2013

Join[{1},Max[#]&/@Table[k^(n-k),{n,25},{k,n}]] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = vecmax(vector(n+1, k, (k-1)^(n-k+1))); \\ Michel Marcus, Jun 13 2017

a(n) = A056155(n-1)^(n - A056155(n-1)), for n >= 2. - Ridouane Oudra, Dec 09 2020

Easdown reference from Michail Kats (KatsMM(AT)info.sgu.ru)

More terms from James Sellers, Aug 21 2000

A265899 After a(1) = 1, positions of descents in A265894.

Original entry on oeis.org

1, 3, 6, 8, 11, 14, 17, 20, 24, 27, 31, 34, 38, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 94, 98, 102, 106, 111, 115, 120, 124, 128, 133, 137, 142, 146, 151, 156, 160, 165, 169, 174, 179, 184, 188, 193, 198, 202, 207, 212, 217, 222, 227, 231, 236, 241, 246, 251, 256, 261, 266, 271, 276, 281, 286, 291, 296, 301

Offset: 1

Antti Karttunen, Dec 24 2015

nonn

Numbers n for which A099563(A001813(n)) <= A099563(A001813(n-1)), where A001813(n) = (2n)! / n!, and A099563 gives the most significant digit in the factorial base representation (A007623) of n.

Antti Karttunen, Table of n, a(n) for n = 1..1503

Cf. A001813, A007623, A099563, A265894.
Cf. A265898 (a subsequence), A266119 (first differences), A266120 (terms immediately before descents).
Cf. also A031435.

A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
A265894 = n -> A099563((2*n)! / n!);
my(i=0, p=1, n=0); while(i < 60, n++; my(k = A265894(n)); if(k <= p, i++; print1(n, ", ")); p = k; );

;; With Antti Karttunen's IntSeq-library.
(define A265899 (MATCHING-POS 1 1 (lambda (n) (<= (A265894 n) (A265894 (- n 1))))))

A307175 Smallest power to which 1+1/n must be raised in order for an interval [k,k+1], with k an integer, to be skipped.

Original entry on oeis.org

4, 7, 9, 13, 15, 18, 22, 26, 27, 32, 33, 40, 42, 48, 51, 55, 58, 62, 66, 71, 75, 80, 85, 85, 91, 97, 103, 105, 111, 112, 120, 121, 129, 131, 139, 142, 143, 153, 156, 158, 168, 172, 175, 178, 181, 193, 197, 201, 206, 210, 215, 220, 225, 230, 235, 241, 246, 252

Offset: 2

Alex Costea, Mar 27 2019

nonn

Here the skipping of an interval means that the interval falls strictly between (1+1/n)^(a(n)-1) and (1+1/n)^a(n).
The sequence is not monotonically increasing; a(24) = a(25) and a(62) > a(63) are the first counterexamples.
Asymptotic to n * log(n), and as such also to the prime numbers (A000040).

			1.1^26 = 11.918... and 1.1^27 = 13.109...; [12,13] is skipped, and this is the first time this happens, thus a(10)=27.

Cf. A031435.

a[n_, m_] := Reduce[(1+1/n)^(m-1) < k < k+1 < (1+1/n)^m, k, Integers];
a[n_] := For[m = 1, True, m++, If[a[n, m] =!= False, Return[m]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jul 07 2019 *)

a(n) = my(k=2, last=1+1/n); while(floor(new = (1+1/n)^k) - ceil(last) != 1, k++; last = new); k; \\ Michel Marcus, Mar 30 2019

from math import floor, log
def get_a_of_n(i):
     x=1+1/i
     j=i
     while floor(log(j, x))!=floor(log(j+1, x)):
         j+=1
     return floor(log(j, x))+1
def main():
     step=1
     i=2
     while True:
         y=get_a_of_n(i)
         print(y, end=", ")
         i+=step

A377206 a(n) = ceiling(log(1/n)/log(1 - 1/n)).

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 16, 19, 22, 26, 29, 33, 36, 40, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 84, 88, 92, 96, 101, 105, 110, 114, 119, 123, 128, 132, 137, 142, 146, 151, 156, 160, 165, 170, 175, 180, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234

Offset: 2

Walter Robinson, Oct 19 2024

This sequence also describes the minimum number of n-player games, where each player has an equal chance of winning, that must be played for a given player to have an equal or greater chance of winning at least once than they have of losing a single game.

Cf. A031435, A094500.

Table[Ceiling[Log[1/n]/Log[1-1/n]], {n,2,58}] (* James C. McMahon, Nov 04 2024 *)

a(n) = A031435(n-1) + 1 for n >= 3.

cp's OEIS Frontend

A003320 a(n) = max_{k=0..n} k^(n-k).

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A265899 After a(1) = 1, positions of descents in A265894.

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A307175 Smallest power to which 1+1/n must be raised in order for an interval [k,k+1], with k an integer, to be skipped.

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A377206 a(n) = ceiling(log(1/n)/log(1 - 1/n)).

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